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PureMathematicsnØêÆ,2021,11(9),1673-1678
PublishedOnlineSeptember2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.119186
Fock˜m¥k.HankelŽfÎÒ
¯¯¯§§§–––’’’¤¤¤
∗
*H“‰ÆêƆÚOÆ§2ÀÖô
ÂvFϵ2021c817F¶¹^Fϵ2021c919F¶uÙFϵ2021c926F
Á‡
éuα
1
,α
2
>0,1<p
2
<p
1
<∞§©•HankelŽfh
α
2
f
lF
p
1
α
1
F
p
2
α
2
k.ž§ÙÎ
Ò¼êfkÛ5Ÿ"
'…c
HankelŽf§Fock˜m§ÎÒ¼ê
TheSymbolFunctionsofBounded
SmallHankelOperators
betweenDifferent
FockSpace
ErminWang,YechengShi
∗
SchoolofMathematicsandStatistics,LingnanNormalUniversity,ZhanjiangGuangdong
Received:Aug.17
th
,2021;accepted:Sep.19
th
,2021;published:Sep.26
th
,2021
∗ÏÕŠö"
©ÙÚ^:¯,–’¤.Fock˜m¥k.HankelŽfÎÒ[J].nØêÆ,2021,11(9):1673-1678.
DOI:10.12677/pm.2021.119186
¯,–’¤
Abstract
Forα
1
,α
2
>0,1<p
2
<p
1
<∞,westudythepropertyofsymbolfunctionsfwhenthe
smallHankeloperatorsh
α
2
f
areboundedfromF
p
1
α
1
toF
p
2
α
2
.
Keywords
SmallHankelOperators,FockSpaces,Symb olFunction
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
C
n
´n‘Em.éC
n
¥?¿ü:z=(z
1
,···,z
n
)Úw=(w
1
,···,w
n
),P
hz,wi= z
1
w
1
+···+z
n
w
n
,|z|=
p
hz,zi.?‰½α>0,•ÄC
n
þGaussVÇÿÝ
dv
α
(z) =

α
π

n
e
−α|z|
2
dv(z),
Ù¥dv(z)´C
n
þIONÈÿÝ.
1 ≤p<∞ž,L
p
α
L«C
n
þ¤k÷v
kfk
p
p,α
=

pα
2π

n
Z
C
n



f(z)e
−
α
2
|z|
2



p
dv(z) <∞
LebesgueŒÿ¼êfN.p= ∞ž,^L
∞
α
L«C
n
þ÷v
kfk
∞,α
= esssup{|f(z)|e
−
α
2
|z|
2
: z∈C
n
}<∞
LebesgueŒÿ¼êfN.w,L
p
α
3k·k
p,α
‰ê¿Âe¤˜‡Banach˜m.
-H(C
n
)L«C
n
þX¼êN,^H
∞
L«C
n
þk.)Û¼êN.½ÂFock
˜m
F
p
α
= L
p
α
\
H(C
n
).
DOI:10.12677/pm.2021.1191861674nØêÆ
¯,–’¤
N´yF
p
α
´L
p
α
4f˜m.Ïd,F
p
α
•´Banach˜m.'uFock˜m˜nØŒ„©
z[1–4].
K
α
(z,w)´F
2
α
2)ؼê,·‚kK
α
(z,w) = e
αhz,wi
,„[3].^k
z
(w) =
K
α
(w,z)
||K
α
(·,z)||
2,α
L
«ÙIOz2)Ø.¯¤±•,lL
2
α
F
2
α
ÝKP
α
ŒL«•
P
α
f(z) =
Z
C
n
K
α
(z,w)f(w)dv
α
(w).
1 ≤p≤∞ž,P
α
þãÈ©Lˆª´lL
p
α
F
p
α
‚5Žf.f∈F
p
α
ž,kP
α
f= f.-
Γ
α
=
(
m
X
j=1
a
j
K
α
(·,z
j
) : m∈N,z
j
∈C
n
anda
j
∈C
)
.
´•Γ
α
´F
p
α
Úf
∞
α
Èf8,Ù¥0 <p<∞.••Bå„,·‚-
F
p
α
= {f: f∈F
p
α
}.
^P
α
L«lL
p
α
F
p
α
ÝK.é?¿a∈C
n
,t
a
(z) = z+a, XJC
n
þLebesgueŒ
ÿ¼êf÷vf◦t
a
∈L
1
(C
n
,dv
α
),K¡fÎÜ^‡(I
1
).w,,fÎÜ^‡(I
1
)…=
Z
C
n
|K
α
(z,a)||f(z)|dv
α
(z) <∞,a∈C
n
.
fÎÜ^‡(I
1
)ž,dfpHankelŽfh
f
Œ3F
p
α
þȽ•
h
f
g(z) = P
α
(fg)(z) =
Z
C
n
K
α
(w,z)f(w)g(w)dv
α
(w).(1.1)
©Ì‡8´ïÄØÓFock˜mƒmk.HankelŽfh
f
ÎÒfäkŸo5
Ÿ.31Ü©,·‚k‰Ñ˜ý•£.31nÜ©,1<p
2
<p
1
<∞ž,·‚y²e
h
α
2
f
: F
p
1
α
1
→F
p
2
α
2
k.,KÙÎÒf7,3˜A½Fock˜mp.
2.ý•£
‰½a∈C
n
,r>0,PB(a,r) = {z∈C
n
: |z−a|<r}.eC
n
¥S{a
k
}÷v:
1)
S
∞
k=1
B(a
k
,r) = C
n
;
2){B(a
k
,
r
4
)}
∞
k=1
p؃.
K¡{a
k
}•C
n
¥˜‡r‚.N´y,?‰½δ>0,o•3Úr,δƒ'êm¦z‡C
n
¥:Ñáu–õm‡8Ü{B(a
k
,δ)}.
‰½r>0,N´Àa
k
∈C
n
,¦{a
k
}´˜‡r‚.
DOI:10.12677/pm.2021.1191861675nØêÆ
¯,–’¤
•y²Ì‡(Ø,·‚I‡Fock˜mf©)½n,„©z[5].
Ún2.1-1 ≤p≤∞.r
0
>0,r÷v0 <r<r
0
,¼êfäkXe/ª:
f(z) =
∞
X
k=1
λ
k
e
αhz,a
k
i−
α
2
|a
k
|
2
,(2.1)
Ù¥{λ
k
}∈l
p
,{a
k
}´˜‡r‚.Kf∈F
p
α
,…•3†fÃ'~êC,¦é?¿f∈F
p
α
,Ñ
k
C
−1
kfk
p,α
≤infk{λ
k
}k
l
p
≤Ckfk
p,α
Ù¥e(.´H¤kŒ±/X(2.1)ª{λ
k
}.
Ùg,·‚‰Ñe¡éó½n,„©z[3].
Ún2.2α,β>0,1 ≤p<∞,q´pÝ•I,=
1
p
+
1
q
= 1.K3(é
hf,gi
γ
=lim
R→∞
Z
|z|<R
f(z)g(z)e
−γ|z|
2
dv(z)
e,F
p
α
éó˜m´F
q
β
,f
∞
α
éó˜m´F
1
β
,Ù¥γ=
√
αβ.
3.k.HankelŽfÎÒ
3ùÜ©,éuα
1
,α
2
>0,1 <p
2
<p
1
<∞,·‚5?ØlF
p
1
α
1
F
p
2
α
2
k.HankelŽf
h
α
2
f
ÎÒ¼êfkÛ5Ÿ.·‚±e½n.
½n3.1.1<p
2
<p
1
<∞,…h
α
2
f
:F
p
1
α
1
→F
p
2
α
2
k..Kf∈F
q
β
,Ù¥q=
p
1
p
2
p
1
−p
2
,β=
α
2
2
α
1
+α
2
.?˜Ú,k
kfk
q,β
≤Ckh
α
2
f
k.
y².ybh
α
2
f
: F
p
1
α
1
→F
p
2
α
2
k..½0 <r<r
0
,Ù¥r
0
X½nA¥‰Ñ,-{a
k
}´˜
‡r‚.Ké?¿{λ
k
}∈l
p
1
,Ún2.1wŠ·‚,¼ê
g
t
(z) =
∞
X
k=1
λ
k
r
k
(t)k
a
k
(z) =
∞
X
k=1
λ
k
r
k
(t)e
α
1
hz,a
k
i−
α
1
2
|a
k
|
2
áuF
p
1
α
1
,…kg
t
k
p
1
,α
1
≤Ck{λ
k
}k
l
p
1
.dh
α
2
f
: F
p
1
α
1
→F
p
2
α
2
´k.,·‚Œ±
kh
α
2
f
g
t
k
p
2
p
2
,α
2
≤kh
α
2
f
k
p
2
·kg
t
k
p
2
p
1
,α
1
≤Ckh
α
2
f
k
p
2
·k{λ
k
}k
l
p
1
.
DOI:10.12677/pm.2021.1191861676nØêÆ
¯,–’¤
Óž,dFubini½nÚKhinchineØªŒ•
Z
1
0
kh
α
2
f
g
t
k
p
2
p
2
,α
2
dt
=
Z
C
n
e
−
p
2
α
2
2
|z|
2
dv(z)
Z
1
0





∞
X
k=1
λ
k
r
k
(t)h
α
2
f
k
a
k
(z)





p
2
dt
≥C
Z
C
n
∞
X
k=1
|λ
k
|
2
|h
α
2
f
k
a
k
(z)|
2
!
p
2
2
e
−
p
2
α
2
2
|z|
2
dv(z)
≥C
∞
X
j=1
Z
B(a
j
,r)
∞
X
k=1
|λ
k
|
2
|h
α
2
f
k
a
k
(z)|
2
!
p
2
2
e
−
p
2
α
2
2
|z|
2
dv(z).
½j,·‚k
∞
X
k=1
|λ
k
|
2
|h
α
2
f
k
a
k
(z)|
2
≥|λ
j
|
2
|h
α
2
f
k
a
j
(z)|
2
.
2(Ü|h
α
2
f
k
a
j
(·)|
q
gNÚ5,k
∞
X
j=1
Z
B(a
j
,r)
∞
X
k=1
|λ
k
|
2
|h
α
2
f
k
a
k
(z)|
2
!
p
2
2
e
−
p
2
α
2
2
|z|
2
dv(z)
≥
∞
X
j=1
|λ
j
|
q
Z
B(a
j
,r)
|h
α
2
f
k
a
j
(z)|
p
2
e
−
p
2
α
2
2
|z|
2
dv(z)
≥C
∞
X
j=1
|λ
j
|
p
2
|h
α
2
f
k
a
j
(a
j
)|
p
2
e
−
p
2
α
2
2
|a
j
|
2
.
q
|h
α
2
f
k
a
j
(a
j
)|=




Z
C
n
f(w)e
α
2
hw,a
j
i
e
α
1
hw,a
j
i−
α
1
2
|a
j
|
2
dv
α
2
(w)




=e
−
α
1
2
|a
j
|
2




Z
C
n
f(w)e
(α
1
+α
2
)hw,a
j
i
dv
α
2
(w)




=: e
−
α
1
2
|a
j
|
2
|J|.
Ïd,
∞
X
j=1
|λ
j
|
p
2
e
−
p
2
2
(α
1
+α
2
)|a
j
|
2
|J|
p
2
≤Ckh
α
2
f
k
p
2
·k{|λ
k
|
p
2
}k
p
2
l
p
2
p
1
.
q
p
1
p
2
Ý•I•
p
1
p
1
−p
2
,ddΥ
n
e
−
p
2
2
(α
1
+α
2
)|a
j
|
2
|J|
p
2
o
∈l
p
1
p
1
−p
2
,…
∞
X
j=1
e
−
q
2
(α
1
+α
2
)|a
j
|
2
|J|
q
≤Ckh
α
2
f
k
q
.
Ù¥q=
p
1
p
2
p
1
−p
2
.eyf∈F
q
β
,Ù¥β=
α
2
2
α
1
+α
2
.-β
0
= α
1
+α
2
,Kβ
0
´βÝ•I.dÚn2.1
DOI:10.12677/pm.2021.1191861677nØêÆ
¯,–’¤
Œ•,é?¿h∈F
q
0
β
0
,Ù¥q
0
´qÝ•I,Ñ•3{µ
j
}∈l
q
0
,Ù¥k{µ
j
}k
l
q
0
≤Ckhk
q
0
,β
0
,¦

h(z) =
∞
X
j=1
µ
j
e
β
0
hz,a
j
i−
β
0
2
|a
j
|
2
.
Ïd,dÚn2.2ÚH¨olderØªŒ
kfk
q,β
'sup
khk
q
0
,β
0
=1
|hh,fi
α
2
|
=Csup
k{µ
j
}k
l
q
0
≤1





Z
C
n
∞
X
j=1
µ
j
e
β
0
hz,a
j
i−
β
0
2
|a
j
|
2
!
f(z)dv
α
2
(z)





=Csup
k{µ
j
}k
l
q
0
≤1





∞
X
j=1
µ
j
e
−
β
0
2
|a
j
|
2
Z
C
n
f(z)e
β
0
hz,a
j
i
dv
α
2
(z)





≤Csup
k{µ
j
}k
l
q
0
≤1
k{µ
j
}k
l
q
0
"
∞
X
j=1
e
−qβ
0
2
|a
j
|
2

Z
C
n
f(z)e
β
0
hz,a
j
i
dv
α
2
(z)

q
#
1
q
≤Ckh
α
2
f
k.
ddŒ•,f∈F
q
β
,…÷vkfk
q,β
≤Ckh
α
2
f
k.½ny.
Ä7‘8
Ø©ÉI[g,‰ÆÄ7“c‘8(12001258),2ÀŽÊÏp“cM#<âa‘8
(2019KQNCX077),*H“‰Æ‰ï‘8(ZL1925)]Ï"
ë•©z
[1]Berger,C.A.andCoburn,L.A.(1987)ToeplitzOperatorsontheSegal-BargmannSpace.
TransactionsoftheAmericanMathematicalSociety,301,813-829.
https://doi.org/10.1090/S0002-9947-1987-0882716-4
[2]Berger,C.A.andCoburn,L.A.(1994)HeatFlowandBerezin-ToeplitzEstimates.American
JournalofMathematics,116,563-590.https://doi.org/10.2307/2374991
[3]Janson,S.,Peetre,J.andRochberg,R.(1987)HankelFormsandtheFockSpace.Revista
Matem´aticaIberoamericana,3,61-138.https://doi.org/10.4171/RMI/46
[4]Tung,J.(2005)FockSpaces.Ph.D.Thesis,UniversityofMichigan,AnnArbor.
[5]Zhu,K.H.(2012)AnalysisonFockSpaces.Springer,NewYork.
DOI:10.12677/pm.2021.1191861678nØêÆ

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